Add & Subtract Fibonacci Numbers: Explained

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1. Can somebody please explain to me how you add Fibonacci numbers using the indices?


2. For example: f(2n+3) + f(2n)
I am actually trying to subtract f(2n) - f(2n+1) - 1
I really have difficulty understanding Fibonacci numbers.
 
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just a correction ...I am trying to subtract f(2n) -( [f(2n+1)] - 1)
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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